61.19.3 problem 3

Internal problem ID [12193]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 3
Date solved : Monday, March 31, 2025 at 04:26:08 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+x f \left (x \right ) y+f \left (x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 49

ode:=diff(y(x),x) = y(x)^2+x*f(x)*y(x)+f(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}}{c_1 -\int {\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}d x}-\frac {1}{x} \]

✓ Mathematica. Time used: 0.709 (sec). Leaf size: 111

ode=D[y[x],x]==y[x]^2+x*f[x]*y[x]+f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\exp \left (-\int _1^x-f(K[1]) K[1]dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-f(K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-f(K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
ode = Eq(-x*f(x)*y(x) - f(x) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*f(x)*y(x) - f(x) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method